Chains of Dense G\( _\delta \) Sets in Perfect Polish Spaces

We prove that in every nonempty perfect Polish space, every dense \( G_\delta \) subset contains strictly decreasing and strictly increasing chains of dense \( G_\delta \) subsets of length \( \mathfrak{c} \), the cardinality of the continuum. As a corollary, this holds in \( \mathbb{R}^n \) for each \( n\ge 1 \). This provides an easy answer to a question of Erdős since the set of Liouville numbers admits a descending chain of cardinality \( \mathfrak{c} \), each member of which has the Erdős property. We also present counterexamples demonstrating that the result fails if either the perfection or the Polishness assumption is omitted. Finally, we show that the set \( \mathcal T \) of real Mahler \( T \)-numbers is a dense Borel set and contains a strictly descending chain of length \( \mathfrak{c} \) of proper dense Borel subsets.